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Title: Geometry, analysis, and morphogenesis: Problems and prospects
The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics, and mathematics. How might these shapes be predicted, and how can they eventually be designed? We review our current understanding of this problem, which brings together analysis, geometry, and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we examine the links between the classical Nash embedding problem and biological morphogenesis. Then, motivated by a range of experimental observations and numerical computations, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the (scaled) thickness becomes vanishingly small and the local curvature can become large. Along the way, we discuss open problems that include those in mathematical modeling and analysis along with questions driven by the allure of being able to tame soft surfaces for applications in science and engineering.  more » « less
Award ID(s):
2011754
PAR ID:
10500031
Author(s) / Creator(s):
;
Publisher / Repository:
Royal Society Publishing
Date Published:
Journal Name:
Bulletin of the American Mathematical Society
Volume:
59
Issue:
3
ISSN:
0273-0979
Page Range / eLocation ID:
331 to 369
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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